A Computational and Experimental Study ofResonators in Three Dimensions

C.K.W. Tam1 and H. Ju2,Florida State University, Tallahassee, FL 32306-4510

M.G. Jones3, W.R. Watson4 and T.L. Parrott5

NASA Langley Research Center, Hampton, VA 23681-2199

In a previous work by the present authors, a computational and experimentalinvestigation of the acoustic properties of two-dimensional slit resonators was carried out.The present paper reports the results of a study extending the previous work to threedimensions. This investigation has two basic objectives. The first is to validate the computedresults from direct numerical simulations of the flow and acoustic fields of slit resonators inthree dimensions by comparing with experimental measurements in a normal incidenceimpedance tube. The second objective is to study the flow physics of resonant linersresponsible for sound wave dissipation. Extensive comparisons are provided betweencomputed and measured acoustic liner properties with both discrete frequency andbroadband sound sources. Good agreements are found over a wide range of frequencies andsound pressure levels. Direct numerical simulation confirms the previous finding in twodimensions that vortex shedding is the dominant dissipation mechanism at high soundpressure intensity. However, it is observed that the behavior of the shed vortices in threedimensions is quite different from those of two dimensions. In three dimensions, the shedvortices tend to evolve into ring (circular in plan form) vortices, even though the slitresonator opening from which the vortices are shed has an aspect ratio of 2.5. Under theexcitation of discrete frequency sound, the shed vortices align themselves into two regularlyspaced vortex trains moving away from the resonator opening in opposite directions. This isdifferent from the chaotic shedding of vortices found in two-dimensional simulations. Theeffect of slit aspect ratio at a fixed porosity is briefly studied. For the range of linersconsidered in this investigation, it is found that the absorption coefficient of a liner increaseswhen the open area of the single slit is subdivided into multiple, smaller slits.

I. Introduction

Local-reacting acoustic liners are one of the most effective devices for suppressing fan noise from modernjet engines. For this reason, acoustic liners are presently installed in almost all commercial aircraft jet engines.Whenever surface area and space are available, engine designers tend to maximize the utilization of acoustic liners.

The structural simplicity of conventional liner structures (i.e., perforate over honeycomb core backed byimpervious backing plate) belies the complexity of their functionality. This makes it difficult to accurately predicttheir acoustic properties. In the past, the impedance of a liner has been determined primarily via semi-empiricalmodels and experiments.1-11 Fairly accurate predictions are possible via semi-empirical prediction models when usedby experienced engineers considering a particular class of liners. However, when a significant change in the linerstructure is made, the prediction of liner impedance becomes less reliable because of the semi-empirical basis of theprediction models. Recent advances in computational aeroacoustics (CAA) offer, for the first time, the potential of afirst-principles prediction of acoustic liner properties. This is most desirable and attractive as the predicted liner

1 Robert O. Lawton Distinguished Professor, Department of Mathematics. Fellow AIAA.2 Research Associate, Department of Mathematics.3 Senior Research Scientist, Structural Acoustics Branch, Research & Technology Directorate. Associate Fellow AIAA.4 Senior Research Scientist, Computational Aerosciences Branch, Research & Technology Directorate. Associate Fellow AIAA.5 Senior Research Scientist, Structural Acoustics Branch, Research & Technology Directorate.

impedance would be free of empirical constants. In addition to the possibility of becoming an impedance predictiontool, CAA offers an enhancement to semi-empirical model prediction by better quantifying constants that heretoforehave necessitated elaborate experimentation, thereby providing a means to investigate and better understand themechanisms by which an acoustic liner dissipates incident sound waves. The openings of an acoustic liner are quitesmall. This makes it difficult to perform experimental observations and measurements of the fluid flow and acousticfields in and around an acoustic liner. On the other hand, small liner openings are no hindrance to CAA simulations.Moreover, numerical simulations using CAA methodologies can provide a complete set of space-time data. This isan invaluable asset for flow physics investigation and analysis. At the present time, the goals of using CAAmethodology for first-principles impedance prediction and for investigation of acoustic wave dissipationmechanisms have not been fully attained. Nevertheless, significant advances have been made. Further efforts arenecessary to develop the technology into a prediction and a design code.

Recently, Tam and Kurbatskii12 initiated an investigation on liner physics and impedance prediction usingdirect numerical simulation (DNS). They recognized that the physics of the problem involved the interplay of theeffects of viscosity and compressibility. Near the walls of the liner opening, the viscous effect is dominant and theviscous scale is small. A thin viscous Stokes layer forms adjacent to the wall. Very fine meshes are required toresolve this Stokes layer. Away from the walls, the dominant effect is compressibility, which gives rise to acousticdisturbances with long wavelengths relative to the slit width. In these regions, the use of a coarse mesh is sufficient.Tam and Kurbatskii took into account the scale difference of the problem and used the multi-size-mesh, multi-time-step, dispersion-relation-preserving (DRP) scheme13 in their numerical simulation. This numerical scheme allows theuse of different size meshes in different regions of the computational domain. They also simplified their problem byconsidering a two-dimensional model. Melling14 provides an excellent historical review of acoustically driven oscillatory flows in square edgedorifices. Starting with Sivian17 in 1935, it has generally been known that such flows exhibit nonlinear acousticresistance wherein the nonlinear increase in resistance is attributed to the irreversible conversion of acoustic energyinto kinetic energy and/or turbulence. Of particular note is the seminal work of Ingard and Labate.9 Via stroboscopicobservations, these researchers discovered four distinct flow regimes (regions in the particle-velocity-frequencyplane) with parametric dependence upon orifice geometry. These regimes were described as follows (in order ofincreasing excitation level) for an orifice over cavity, driven at resonance: Regime 1: A low sound intensity regime with stationary circulation; flow directed out from orifice along axis. Regime 2: A regime of stationary circulation in which the direction of flow along the axis is toward the orifice. Regime 3: A medium sound intensity regime where pulsatory effects are superposed upon circulations of the kind

in Regime 2. Regime 4: A high sound intensity regime in which pulsatory effects are predominant, resulting in the formation of

jets and vortex rings. The jet consists of strong air-flow through the orifice signified by a sudden burst of air. This burst appears symmetrically on both sides of the orifice and is made up of pulsescontributed by each cycle of the sound wave.

It is clear that the quantitative results of the present investigation are consistent with the qualitative results attributedto Regime 4 as described by Ingard, et al.9 The current investigation (beginning with the work of Tam andKurbatskii12) is believed to represent the first quantification of these physical mechanisms via a first-principles CAAmethodology. As such, this represents a significant advance for the understanding of oscillatory, orifice flows ofinterest in the predictive modeling of nonlinear acoustic resistance.

Experimental support for the vortex shedding dissipation mechanism is provided in the work of Tam,Kurbatskii, Ahuja and Gaeta.15 In their experiment, laser visualization was used to scan the unsteady flow fieldinside the resonator. Evidence of the observation of shed vortices was reported in their work. In addition, goodagreement was found between absorption coefficients measured by normal incidence impedance tube experimentand direct numerical simulation. In spite of the good agreements found, experience with computational fluiddynamics (CFD) suggests that a more thorough validation of numerical simulation results is required. In an effort toprovide further support of the validity of the results of direct numerical simulation of acoustic liners, a collaborativestudy was undertaken between the NASA Langley Research Center liner physics team and a Florida StateUniversity (FSU) team. Under this joint effort, the NASA team performed a series of normal incidence impedanceexperiments and measured the impedance of six specimens over the frequency range of 500 to 3000 Hz and SPLs of120 to 155 dB. The FSU team was responsible for conducting a corresponding series of virtual experiments usingDNS. This being a first step in the validation process, two-dimensional (aspect ratio of 40) slit models were used.The extensive results of this joint study have been reported in Tam et al.16 Over the fairly large test parameter space,Tam et al showed that the computed liner impedance was in good agreement with experimental measurements. The

good agreements lend strong support to the validity of the DNS approach, and provide verification that vortexshedding is the main dissipation mechanism of acoustic liners at high SPLs.

Figure 1. Sketch of a line vortex and a ring vortex

The present study is a continuation of the work reported by Tam et al.16 The primary objective is to providethree-dimensional validation tests for direct numerical simulation of the acoustic and flow fields inside a normalincidence impedance tube. Both discrete and broadband incident sound waves are considered. An improved methodis introduced in this work. A secondary objective of this investigation is to clarify the behavior of the shed vortices.For slit resonators with large aspect ratios, the shed vortices are essentially line vortices as shown in Fig. 1. Previouswork indicates that the movements of line vortices, once shed, are quite random and chaotic. However, for acousticliners with an aspect ratio of the face sheet openings close to unity (i.e., slits shrinking toward squares), the shedvortices are in the form of ring vortices. Ring vortices have strong self and mutual interaction. It is our intention toobserve their behavior using the DNS data and to report some of the salient features here.

The remainder of this paper is organized as follows. The experimental procedure and instrumentation arereported in section II. In section III, the three-dimensional computational model and computational algorithm aredescribed. Treatment and analysis of simulation data for broadband incident sound waves are considered in SectionIV. Numerical results and their comparisons with experimental measurements are reported in section V. Someinteresting observations of vortex shedding and behavior of shed vortices are reported in Section VI. Section VIIconcludes this study.

II. Experimental Instrumentation and Procedure

In the present study, experimental data were acquired in a normal incidence impedance tube (NIT) locatedin the Flow Impedance Test Laboratory of the NASA Langley Research Center. A schematic diagram of theapparatus and associated electronics is shown in Fig. 2. The impedance tube has a square cross-section of 2 in by 2in and a length of 24 in. A detailed description of the apparatus may be found in Ref.[16] and will, therefore, not berepeated here. The two-microphone method (TMM) of Jones and Parrott11 is implemented with two microphonesflush-mounted 1.25 in apart in a rotating steel plate in the top wall. This allows the position of two 0.25-in diametercondenser-type microphones to be switched in a very convenient and precise manner. The acoustic impedance of thetest specimen, determined using the TMM, can then be used to separate the total acoustic pressure into incident andreflected acoustic pressures.

Figure 2. NASA LaRC normal incidence tube with instrumentation.

Six test specimens, all with the same open area, were manufactured. However, only data from three of thesix specimens are used to compare with direct numerical simulation (DNS) measurements. We will refer to them asSamples 1, 5 and 6. The plan views of these specimens are shown in Fig. 3.

Figure 3. Slit configurations for test Samples 1, 5 and 6, with aspect ratios of 40, 2.5 and 1.25, respectively.

Sample 1 has a single 0.05 in by 2.0 in slit. Sample 5 has sixteen 0.05 in by 0.125 in slits. Sample 6 hasthirty-two 0.05 in by 0.0625 in slits. Thus, the aspect ratios of the slits are 40, 2.5 and 1.25, respectively. The slit inSample 1 is effectively two-dimensional. The data from this specimen has been used to compare with DNSmeasurements in Ref. [16]. For this reason, the present investigation concentrates on experimental data from theother two samples. With identical total open area, the three samples offer an opportunity to study the effect of holesize on liner impedance.

III. Computational Model, Algorithm and Grid

One of the principal objectives of this investigation is to perform direct numerical simulation of the normalincidence impedance tube experiments carried out at the NASA Langley Research Center. In this section, thecomputational model is presented. In order to have an accurate numerical simulation, a well-designed computationalgrid and a high resolution computational algorithm are required. The three-dimensional computational grid andtime-marching algorithm used in this study are described below.

A. Mathematical Model

The acoustic field inside a normal incidence impedance tube is governed by the compressible Navier-Stokes equations. With respect to the following scales [length = D (thickness of resonator opening), velocity =

a0 (speed of sound), time = D / a0 , density = r0 (mean density of gas), pressure = r0a02 , impedance = r0a0 ] the

dimensionless Navier-Stokes equations are,

∂r∂t

+ r∂u j

∂x j

+ u j∂r∂x j

= 0 , (1)

∂ui

∂t+ u j

∂ui

∂x j

= -1r

∂p∂xi

+1r

∂t ij

∂x j

, (2)

∂p∂t

+ u j∂p∂x j

+ g p∂u j

∂x j

= 0 , (3)

where

t ij =1R

∂ui

∂x j

+∂u j

∂xi

Ê

ËÁ

ˆ

¯˜

. (4)

R =Da0

n is the Reynolds number and g is the ratio of specific heats. In this problem, there is no intrinsic

characteristic velocity (among the input variables) other than the speed of sound. Thus, the Reynolds number, unlikestandard fluid dynamics problems, is based on sound speed. Computationally, the choice of velocity scale makeslittle difference.

Figure 4. Computational domain for a one-slit model.

The Sample 5 and 6 specimens used in the NASA experiment have many openings in the face sheet (seeFig. 3). Computationally, it is not feasible to grid all the openings to reproduce the experimental condition preciselyas this will require enormous number of mesh points and long computation time. To render the computationrequirements reasonable, we will assume that the array of slits is infinite in extent. Under this condition, the acousticfield is spatially periodic. It is then sufficient to perform a computation for one slit with the imposition of periodicboundary conditions. Fig. 4 shows the plan view of the computational domain for a single slit. Periodic boundaryconditions are imposed on vertical surfaces AD and BC as well as on vertical surfaces AB and DC. Thecomputational model in three dimensions is shown in Fig. 5. The dimensions are the same as in the NASAexperiment. To ensure that the computed results of the single slit model are valid, a two-slit model with periodicboundary conditions as shown in Fig. 6 is also developed. It is found that the two-slit model gives essentially thesame results as the one slit model. While this indicates the single-slit model is satisfactorily capturing the multi-slitphsyics, it also indicates the absence of interaction between adjacent holes. This is not altogether surprising, sincethe slit separation-to-slit diameter ratio is fairly large. This two-slit model offers the potential for exploration of thishole interaction effect as the hole separation is reduced. However, there was insufficient time to thoroughlyinvestigate this effect in the current study.

Figure 5. Three-dimensional, single-slit computational model.

Figure 6. A two-slit computational model.

B. Grid Design and Computational Algorithm

It was pointed out in our previous work, Ref. [16], that the normal incidence impedance tube problem is amulti-scale problem. Near the walls of the slit, the viscous effect is dominant. The relevant length scale, L, is givenby (see White18),

L = 2 pnf

Ê

ËÁˆ

¯̃

12

(5)

where f is the frequency of oscillation. Away from the faceplate, the dominant effect on the dynamics of motion iscompressibility. The length scale is the acoustic wavelength, l , which is given by,

l =a0

f. (6)

It is straightforward to show that l / L is equal to 320 for 6 kHz sound at standard conditions.

Because of the large disparity in length scales, a multi-size mesh is adopted. The time-marchingcomputation is carried out using the multi-size-mesh, multi-time-step, dispersion-relation-preserving (DRP) schemeof Ref. [13]. The mesh sizes are chosen so that there are at least seven mesh points per wavelength. Seven meshpoints per wavelength is the accuracy requirement of the 7-point stencil DRP scheme19. Fig. 7 shows the grid designin the x-z plane for the Sample 6 computation. The smallest size mesh is in the sub-domain containing the slitopening. Here, the mesh size D is equal to 0.833 ¥ 10-3 in. The grid design in the x-y plane (the smaller side of theslit) is shown in Fig. 8. It is to be noted that the mesh size in the sub-domain right at the slit opening is the same inthe x, y, and z directions. In both Fig. 7 and 8, the mesh size Dk means it is k times that of D , i.e. Dk = kD . The

number preceding Dk is the number of mesh points of that size being used in that sub-domain.

For a viscous fluid, the boundary condition at a solid wall is the no-slip boundary condition. However, theviscous effect is unimportant away from the slit opening. To save computation time, the no-slip boundary conditionis imposed only on the wall boundaries of the three sub-domains with the smallest size mesh. Effectively, the Eulerequations are used for time-marching computation in the other sub-domains. The wall boundary conditions areenforced by the ghost point method as elaborated in Ref. [16].

Figure 7. Mesh distribution in the x-z plane. Figure 8. Mesh distribution in the x-y plane.

IV. Broadband Incident Acoustic Waves

Acoustic liners are not only effective in suppressing tones. They can also be used to suppress broadbandnoise. One objective of the present investigation is to perform experimental measurements and direct numericalsimulations with a broadband source. In our previous work on two-dimensional slit resonators, Ref. [16], broadbandincident sound was considered. However, the accuracy of the proposed method for analyzing the simulation datawas restricted to an incident sound field with a relatively narrow bandwidth. In this work, a much improved methodis devised that has almost no bandwidth restriction. While this improved method allows modeling of broadband

sound with a large bandwidth, it does not alleviate the need for sufficient sample length. It is well known thatspurious oscillations can be introduced into the frequency spectrum if the input data sample is of insufficientduration. Due to CPU time restrictions, this problem persists.

A. Energy Conserving Discretization

The first crucial requirement in performing numerical simulation of broadband sound is the development ofa method to computationally reproduce the experimentally measured noise spectrum. This is not a trivial problem. Inour previous work, a way to discretize the input noise spectrum so as to reproduce the random noise signal in thetime domain was described. This method of discretization is energy conserving. That is, the energy contained in agiven frequency band of the noise spectrum in the computation is the same as that in the experimentally measuredspectrum.

Figure 9. Energy conserving discretization of a given spectrum.

The energy conserving discretization method divides the measured noise spectrum into a large number ofnarrow bands with unequal bandwidth as show in Fig. 9. One important specification is that no center frequency ofany frequency band is an integral multiple of the center frequency of another band. Harmonic interaction may bepresent in the measurement process due to the nonlinearity of the acoustic drivers. However, for the computationalmodeling, it is important that the discretization method avoid harmonic interaction, and the current schemeaccomplishes this purpose. Suppose Dw j and w j are the bandwidth and center frequency (w j = 2p f j is the

angular frequency) of the j th band, then the broadband sound field is mathematically represented by,

p(t) = 2Dw jS j( )12 cos w jt + c j( )

j =1

N

 (7)

where Sj is the spectrum level at w j and c j is a randomly assigned number. In the Appendix, it is shown that

equation (7) is, indeed, an energy conserving discretization of a given spectrum S w( ) in the time domain. Fig.10 is

the time history of an input broadband sound field generated by equation (7). The spectrum is from one of theNASA Langley Research Center experiments during the present investigation. The time history of pressurefluctuations in Fig. 10 resembles those measured experimentally.

Figure 10. Time history of pressure fluctuations computed according to Eq. (7).

B. Computation of Impedance and Absorption Coefficient

The standard practice of measuring the normal incidence acoustic impedance (Note: all impedances usedherein are assumed to be normalized by the characteristic impedance of air, r0a0 ) of a liner specimen in a normal

incidence impedance tube is to use a two-microphone method as shown in Fig. 11. In our previous work, Ref. [16], away to analyze the output from the simulated microphone signals for determining the impedance of the linespecimen was discussed. However, it was noted that the method was subjected to certain limitations. As a part of thepresent investigation, an improved method has been developed. This method is described below.

It is assumed that only plane waves are present in the normal incidence impedance tube away from thesurface of the liner. Without loss of generality, the pressure and velocity fields inside the tube may be written in theform,

p(x,t) = A w( )eiw x + B(w )e- iw xÈÎ ˘̊e- iw tdw-•

•

Ú , (8)

u(x,t) = A(w )eiw x - B(w )e- iw xÈÎ ˘̊e- iw tdw-•

•

Ú , (9)

where A w( ) and B w( ) are the amplitude functions of the reflected and incident waves. Let Z w( ) be the

impedance of the resonator. The impedance boundary condition at x = 0 is,

%p w( ) = -Z w( ) %u w( ) , (10)

where ~ indicates Fourier transform and an e- iw t time dependence is assumed. By definition,

%p w( ) and %u w( ) are

equal to the quantities inside the square brackets of Eqs. (8) and (9). Upon substitution into Eq. (10), it is easy tofind,

Z w( ) =A w( ) + B(w )B(w ) - A(w )

. (11)

Figure 11. Schematic diagram of a normal incidence impedance tube showing incident and reflected plane waves and the locations of microphone 1 and 2.

Now, let T1(t) be the output of microphone 1 at x1 and T2 (t) be that of microphone 2 at x2 (see Fig. 11).

The Fourier transforms of these outputs are,

%T1(w ) =1

2pT1(t)e

iw tdt ; %T2 (w ) =1

2pT2 (t)eiw tdt

-•

•

Ú-•

•

Ú . (12)

But T1(t) and T2 (t) are given by Eq. (8) with x = x1 and x2 respectively. It follows that

%T1(w ) = A(w )eiw x1 + B(w )e- iw x1 , (13)

%T2 (w ) = A(w )eiw x2 + B(w )e- iw x2 . (14)

It is straightforward to solve Eqs. (13) and (14) for A(w ) and B(w ) . Upon substitution into Eq. (11), it is easy to

obtain,

Z(w ) = i%T1(w )sin(w x2 ) - %T2 sin(w x1)%T1(w )cos(w x2 ) - %T2 cos(w x1)

È

ÎÍ

˘

˚˙

. (15)

Once the normalized normal incidence impedance Z(w ) is found by Eq. (15), the corresponding normalized

reflection factor and absorption coefficient may be found by

Ref Factor =

†

Z(w) -1Z(w) +1

, Absorption Coefficient = 1-Z(w ) - 1.0Z(w ) + 1.0

2

. (16)

V. Numerical Results and Comparisons with Experiments

For this investigation, we concentrate our effort on experimental validation of DNS results for TestSamples 5 and 6. (Results of the validation tests for Sample 1 were previously reported in Ref. [16].) Two types oftests are performed in the present study - one with discrete frequency incident sound and the other with broadbandacoustic waves. Since the response of a resonator to discrete frequency sound is quite different from that ofbroadband noise, the results are reported separately below.

A. Discrete Frequency Incident Sound

Sample 6 (32 slits) results

Two test series were performed with Sample 6, in which the source frequency was held constant and theincident sound pressure level (SPL) was varied. The first series was conducted at 2000 Hz, and the second wasconducted at 2500 Hz. These frequencies were chosen for potential comparison with earlier results based on two-dimensional DNS computations.

For the test series conducted at 2000 Hz, the incident sound pressure level (SPL) was varied from 114 dBto 145 dB. This variation in SPL level is 31 dB, which is quite substantial. Figs. 12 and 13 show the measured andcomputed liner impedance (resistance and reactance) as a function of incident sound pressure level. In all thesimulations, the pressure output of a pair of microphones located at 2.5 and 3.5 in from the surface of the resonatorare used for calculating the impedance. These are the same as the locations of the microphones used in theexperiment. To provide a direct check on the accuracy of the simulation, the measured signals of a second pair ofmicrophones located at 10 and 11 in from the surface of the resonator are also used. We have found in all the testcases that the two pairs of microphones give virtually identical resonator impedance. Hence, only the results for thepair of microphones located at 2.5 and 3.5 in from the resonator surface are shown. The measured normalizedresistance is nearly 0.35 (recall that impedances in this paper are normalized by r0a0 ) below the computed value at

the lowest incident SPL, but this separation is reduced as the incident SPL is increased. The measured and computedreactance differences are larger, with a consistent difference of nearly 0.8 between the two sets of results over theentire range of incident SPLs.

Figure 12. Normalized acoustic resistance at different sound pressure levels with 32-slit sample;

incident sound frequency of 2 kHz. u, experiment; ¢, simulation.

Figure 13. Normalized acoustic reactance at different sound pressure levels with 32-slit sample;

incident sound frequency of 2 kHz. u, experiment; ¢, simulation.

These impedance differences between the DNS and experimental results are clearly of concern. However, itshould be noted that cavity anti-resonances occur at 1125 and 2250 Hz. When combined with the mass reactance ofthe liner, this will result in anti-resonant frequencies for the entire resonator (facesheet plus cavity) that are belowthose of the cavity alone. Hence, the choice of 2000 Hz is problematic, and represents one of the more difficult casesfor which to make these comparisons. Nevertheless, it provides a worthy goal to demonstrate eventual success.

A better understanding of these differences emerges when the data are replotted in a different format. Figs.14 and 15 provide the corresponding measured and computed normalized reflection factor (magnitude and phase)spectra. Figure 14 shows the reflection factor magnitude to be well matched for all but the highest SPL, with only aslight degradation in the comparison at this SPL. Figure 15 shows the measured reflection factor phase to beconsistently higher than that computed via DNS. The difference is approximately 4 degrees for all but the highestSPL, and decreases to 2 degrees at the highest SPL. Hence, at least for this test condition, it appears that thedifferences in the measured and computed impedances are likely the result of a problem associated with the phase.

There are at least three possible causes for this phase difference. First, the DNS simulations were allconducted at a constant sound speed of 340 m/s (assumed standard day conditions), whereas the sound speed in theexperiments was approximately 345 m/s. This difference in sound speeds will cause a constant phase shift betweenthe simulation and the measurement, and is of the appropriate magnitude to result in the phase shift observed infigure 15. However, as will be shown for other test conditions, the phase difference is not always constant. Thus, itwould appear that this difference in sound speeds is not the sole cause of differences between the simulations andmeasurements. Regardless, this effect should be included in future computations. Second, measurement of incidentSPL at the surface of the resonator requires accurate measurements, especially for frequencies near anti-resonance.Thus, it may be necessary to revisit the experiment with more detailed measurements. Finally, the computationaldomain selected for the DNS simulations may cause this phase difference. These simulations do not account for theeffects of the walls of the normal incidence tube, as they are currently assumed to be sufficiently far from theresonator opening such that they can be neglected. However, near the surface of a nonlinear liner, higher-ordermodes are set up between the opposite walls of the normal incidence tube. These higher-order modes die out beforethey get to the location of the microphones, but they influence the near-field effect of the resonator openings. It istherefore possible that neglecting this effect may result in a phase difference in the results computed at themicrophone locations. Each of these possibilities should be explored in future investigations.

Figure 14. Normalized reflection factor magnitude at different sound pressure levels with 32-slit sample;

incident sound frequency of 2 kHz. u, experiment; ¢, simulation.

Figure 15. Normalized reflection factor phase at different sound pressure levels with 32-slit sample;

incident sound frequency of 2 kHz. u, experiment; ¢, simulation.

Figure 16. Absorption coefficient at different sound pressure levels with 32-slit sample;

incident sound frequency of 2 kHz. u, experiment; ¢, simulation.

Figure 16 provides the corresponding absorption coefficient spectra. Although the absorption is quite low,there is quite good agreement between the measured and computed results. Thus, for a normal incidence tubeenvironment, the differences in the measured and computed impedances are not significant. However, the reader isreminded that the absorption coefficient is not an intrinsic parameter of the liner, and instead only provides

information regarding how the liner will respond in the selected environment. Thus, since we wish to eventually usethe DNS approach to model liners used in actual aircraft engine environments, comparisons of the measured andcomputed liner impedances (or alternatively, the reflection factor) remain as the appropriate figure of merit.

Figure 17. Normalized acoustic resistance at different sound pressure levels with 32-slit sample;

incident sound frequency of 2.5 kHz. u, experiment; ¢, simulation.

Figure 18. Normalized acoustic reactance at different sound pressure levels with 32-slit sample;

incident sound frequency of 2.5 kHz. u, experiment; ¢, simulation.

Figure 19. Normalized reflection factor magnitude at different sound pressure levels with 32-slit sample;

incident sound frequency of 2.5 kHz. u, experiment; ¢, simulation.

Figure 20. Normalized reflection factor phase at different sound pressure levels with 32-slit sample;

incident sound frequency of 2.5 kHz. u, experiment; ¢, simulation.

Figure 21. Absorption coefficient at different sound pressure levels with 32-slit sample;

incident sound frequency of 2.5 kHz. u, experiment; ¢, simulation.

A similar series of validation tests was performed at a sound frequency of 2500 Hz. Figs. 17 – 21 providethe measured and computed liner impedance (resistance and reactance), reflection factor (magnitude and phase), andabsorption coefficient as a function of incident sound pressure level. The comparisons of measured and computedimpedance (resistance and reactance) are improved for this frequency at the lower incident SPLs, but degrade at thehigher SPLs. The reflection factor (magnitude and phase) and absorption coefficient comparisons follow similartrends, again indicating at least a portion of the disparity between measured and computed results are due to a phasemismatch.

Figure 22. Normalized acoustic resistance at different frequencies with 32-slit sample;

incident sound pressure level of 146 dB. u, experiment; ¢, simulation.

Figure 23. Normalized acoustic reactance at different frequencies with 32-slit sample; incident sound pressure level of 146 dB. u, experiment; ¢, simulation.

Figure 24. Normalized reflection factor magnitude at different frequencies with 32-slit sample;

incident sound pressure level of 146 dB. u, experiment; ¢, simulation.

Figure 25. Normalized reflection factor phase at different frequencies with 32-slit sample;

incident sound pressure level of 146 dB. u, experiment; ¢, simulation.

Figure 26. Normalized reflection factor phase at different frequencies with 32-slit sample;

incident sound pressure level of 146 dB. u, experiment; ¢, simulation.

Finally, data were acquired with Sample 6 for a variety of frequencies, with the incident SPL held constantat 146 dB. It is evident from Fig. 23 (reactance) that the cavity resonance is the dominant factor (-cot(kL) behavior).In general, the comparison of measured and computed resistance is quite acceptable, except at 2500 Hz. However,the differences between measured and computed reactances are greater than 0.5 for most of the test frequencies.Again, a review of the reflection factor magnitude and phase results (Figs. 24 and 25) provides insight. Themagnitudes are matched to within 0.1 for all but 2500 Hz, yet the phase differences hover at or above 5 degrees(interestingly, smaller at 2500 Hz). Clearly, this phase mismatch must be addressed in further investigations.

Sample 5 (16 slits) results

Two test series were performed for the 16-slit sample. First, the frequency was held constant at 2 kHz andthe incident SPL was varied. Next, the incident SPL was held constant at 150 dB and the frequency was varied from500 to 3000 Hz. In general, the comparisons achieved with this sample were very similar to those achieved with the32-slit sample.

Figure 27. Normalized acoustic resistance at different sound pressure levels with 16-slit sample;

incident sound frequency of 2 kHz. u, experiment; ¢, simulation.

Figure 28. Normalized acoustic reactance at different sound pressure level with 16-slit sample;

incident sound frequency of 2 kHz. u, experiment; ¢, simulation.

Figure 29. Normalized reflection factor magnitude at different sound pressure levels with 16-slit sample; incident sound frequency of 2 kHz. u, experiment; ¢, simulation.

Figure 30. Normalized reflection factor phase at different sound pressure levels with 16-slit sample;

incident sound frequency of 2 kHz. u, experiment; ¢, simulation.

Figure 31. Absorption coefficient at different sound pressure levels with 16-slit sample;

incident sound frequency of 2 kHz. u, experiment; ¢, simulation.

Figure 32. Normalized acoustic resistance at different frequencies with 16-slit sample;

incident sound pressure level of 150 dB. u, experiment; ¢, simulation.

Figure 33. Normalized acoustic reactance at different frequencies with 16-slit sample;

incident sound pressure level of 150 dB. u, experiment; ¢, simulation.

Figure 34. Normalized reflection factor magnitude at different frequencies with 16-slit sample;

incident sound pressure level of 150 dB. u, experiment; ¢, simulation.

Figure 35. Normalized reflection factor phase at different frequencies with 16-slit sample;

incident sound pressure level of 150 dB. u, experiment; ¢, simulation.

Figure 36. Absorption coefficient at different frequencies with 16-slit sample;

incident sound pressure level of 150 dB. u, experiment; ¢, simulation.

B. Broadband Incident Acoustic Waves

Two normal incidence impedance tube experiments were conducted using broadband incident soundwaves. In these experiments, Sample 6 was used as the face sheet of the resonator. Because of reflection from theface sheet of the resonator, it is very difficult to produce incident acoustic waves with a prescribed spectrumexperimentally. For the two experiments reported below, the acoustic drivers were used to generate a broadbandspectrum. The acoustic field inside the impedance tube was measured by the two-microphone method. The incidentsound wave spectrum was then deduced by analyzing the output of the two microphones. This spectrum was thenreproduced by the energy conserving discretization method, discussed in an earlier section, for use as input in directnumerical simulation.

Figure 37. Incident broadband noise spectrum at 141 dB OASPL from the NASA experiment.

Fig. 37 shows the measured incident broadband noise spectrum at 141 OASPL (dotted line). The spectrumcovers the range of 500 Hz to 3000 Hz. In the numerical simulation, the spectrum was divided into 118 bands. Thecircles in Fig. 37 are the center frequencies of the bands. As can be seen the discretized spectrum (full line) is a veryclose approximation of that of the experiment.

Figure 38. Comparison of measured and computed resistance spectra. OASPL = 141 dB. Full line – experiment. Dotted line – numerical simulation.

Figure 39. Comparison of measured and computed reactance spectra. OASPL = 141 dB. Full line – experiment. Dotted line – numerical simulation.

Figs. 38 and 39 show the measured resistance and reactance (full line) in the frequency range of 500 Hz to3000 Hz. The computed resistance and reactance spectra (dotted line) are also shown. As can be seen, the computedand experimental results follow very similar trends. The numerical simulation reasonably captures the resistancespike that occurs at the first anti-resonance. Similarly, the large drop in reactance that occurs at this frequency is alsocaptured (see Fig. 39). However, the second anti-resonance near 2200 Hz is not properly captured. The authors havenot yet found the source of this disparity. However, the agreement between measured and computed impedances isimproved away from these anti-resonance frequencies. It should be noted that they differ by at least 1.0 at somefrequencies. Regardless, the overall comparison is quite good, especially given the complexity of the computationalprocess.

Figure 40. Comparison of measured and computed absorption coefficient spectra. OASPL = 141 dB. Full line – experiment. Dotted line – numerical simulation.

Fig. 40 shows a comparison between the measured and computed absorption coefficient spectra. There isreasonable agreement up to 2700 Hz. The measured data is relatively smooth. In contrast, the computed curve isjagged. It is believed that the sample length of the computed data used to compute the absorption coefficient is onlymodestly long. Had a longer sample length been simulated, the result would be a much smoother curve. For futureinvestigations, it is also suggested that multiple simulations with extended length be conducted, such that theindividual spectra are smoother and the uncertainty associated with the simulation of broadband data can beevaluated statistically.

Figure 41. Incident broadband noise spectrum at 150.4 dB OASPL from the NASA experiment.

The same experiment and numerical simulation was repeated at a higher incident sound pressure level. Fig.41 shows the measured incident sound spectrum at an OASPL of 150.4 dB (dotted line). The spectrum is dividedinto 117 bands for numerical simulation. The band center frequencies are indicated by circles in this figure. Figures42, 43 and 44 show comparisons of the resonator resistance, reactance and absorption coefficient spectra betweenexperimental measurement and numerical simulation results. The agreements are similar to those at 141 dB OASPL.In this case, the numerical results of reactance, Fig. 43, have two anti-resonances at about the same frequencies as

the experiment. The peak levels are, however, a bit low. Other than the anti-resonances, there is fairly goodagreement over the entire frequency range considered.

Figure 42. Comparison of measured and computed resistance spectra. OASPL = 150.4 dB. Full line – experiment. Dotted line – numerical simulation.

Figure 43. Comparison of measured and computed reactance spectra. OASPL = 150.4 dB. Full line – experiment. Dotted line – numerical simulation.

Figure 44. Comparison of measured and computed absorption coefficient spectra. OASPL = 150.4 dB. Full line – experiment. Dotted line – numerical simulation.

C. Effect of Slit Aspect Ratio

By keeping the porosity ratio and slit width fixed, it is clear that an increase in the number of slits wouldlead to a decrease in the aspect ratio. On the other hand, the total length of the perimeter of the slit openings wouldincrease. This, invariably, would have an impact on the acoustic performance of a resonant liner. In this study,computed results are available at slit aspect ratio of 40 (Sample 1), 2.5 (Sample 5) and 1.25 (Sample 6). Figs. 45 and46 show the change in resistance and reactance over the SPL range of 114 dB to 145 dB due to a change in slitaspect ratio at a frequency of 2000 Hz. It is clear that a decrease in the aspect ratio (i.e., distributing the open areaamong an increased number of slits) results in an increase in resistance and a decrease in reactance over the entireSPL range included in this investigation. Fig. 47 shows the corresponding increase in absorption coefficient. At thelow SPL range of the study (around 114 dB), there is no vortex shedding, and the damping of incident acousticswaves is largely due to wall friction. As the open area is subdivided into a larger number of small slits, the totalperimeter of all the slits is larger than that of the individual large slit. Thus, one would expect larger dissipation andhence an increase in absorption coefficient with longer perimeter of slit openings.

At high SPL, the dominant dissipation mechanism is vortex shedding. This causes the resistance of thesingle slit (aspect ratio of 40) to increase at this SPL. As the slits become smaller, the sample thickness-to-slit widthratio increases. Under this condition, the viscous dissipation due to the wall friction increases. When combined withthe dominant vortex shedding at high SPL, this results in further resistance increases, which leads to increasedabsorption coefficient.

Figure 45. Variation of liner resistance with slit aspect ratio at 2000 Hz frequency.

u , p, ¢ slit aspect ratio, 40, 2.5, 1.25.

Figure 46. Variation of liner reactance with slit aspect ratio at 2000 Hz frequency.

u , p, ¢ slit aspect ratio, 40, 2.5, 1.25.

Figure 47. Variation of liner absorption coefficient with slit aspect ratio at 2000 Hz frequency. u , p, ¢ slit aspect ratio, 40, 2.5, 1.25.

VI. Vortex Shedding and Behavior of Shed Vortices

One advantage of numerical simulation is that it provides a full set of spatial and temporal data. We makeuse of this data to observe the vortex-shedding phenomenon at the resonator opening. For convenience ofobservation, a number of videos on the fluid motion in the vicinity of the resonator are made. They show the motionof the vortices in different planes that cut through the normal incidence impedance tube. Vortex motion under theexcitation of single frequency incident sound waves and broadband incident sound waves are recorded and analyzed.A short summary of the observed vortex behavior is provided below.

A. Single Frequency Incident Sound Waves

When a vortex is shed, its plan form initially has the same shape as the resonator opening. It is not circular.However, after the vortex moves away from the resonator opening, it becomes a relatively free vortex. It is observedthat such a vortex has the tendency to adjust itself to become circular. The large dimension of the vortex ringgradually reduces. This can be seen in Fig. 48. This figure shows the density distribution in a plane cutting the largerwidth of the slit (Sample 5). In this figure, the width of the vortex farther away from the resonator opening is smallerthan that of the vortex just shed. On the other hand, the small dimension of the vortex ring increases. This is shownin Fig. 49. This figure corresponds to that of a plane cutting through the smaller dimension of the slit. It is easy toobserve in this figure that the width of the vortex farther away from the resonator opening is larger than that of thevortex just shed. At this time, it is not clear why a free vortex tends to become circular. It is possible that a circularvortex is the natural stable configuration.

Figure 48. Vortex motion as viewed in a plane Figure 49. Vortex motion as viewed in a plane cutting through the larger side of the slit opening. cutting through the smaller side of the slit opening.

The shed vortices form two vortex trains one on each side of the resonator opening. The vortices maintain afairly regular spacing (see Fig. 50). Both vortex trains move away from the resonator opening in opposite directions.It appears that the vortices move in formation due to mutual interaction. The directions of rotation of the vortices onopposite sides of the resonator opening are opposite. This is believed to be the principal reason why the vortex trainsmove in directions opposite to each other.

Figure 50. Instantaneous density distribution showing two vortex trains shed from a resonator opening.

B. Broadband Incident Sound Waves

When a slit resonator is subjected to excitation by broadband incident sound waves at high SPL, the flowfield around the resonator opening is again dominated by vortex shedding. However, vortices are shed randomly(see Fig. 51). In some cases, two vortices are shed on the same side of the resonator opening before a vortex is shedon the other side (see Fig. 52). Vortex shedding is a nonlinear phenomenon. Thus, we should not expect the acousticcharacteristics of a liner at a given frequency in the presence of broadband noise to be the same as that of the liner

under discrete frequency forcing. In other words, if Z w( ) is the resonator impedance under broadband incident

sound and Zw is the impedance of the resonator under a single frequency incident sound wave at the same

frequency w , then, in general Z(w ) ≠ Zw .

`

Figure 51. Random vortex shedding under excitation Figure 52. Non-periodic random vortex shedding. by broadband incident sound waves.

VII. Conclusion and Summary

One of the principal objectives of the present investigation is to validate, by experimental measurements,three-dimensional direct numerical simulation results of slit resonators in a normal incidence impedance tube. Bothdiscrete and broadband frequency incident sound waves are used. Test cases include slits with substantially differentaspect ratios. It has been found the simulation results track the trends of the measured impedances quite well, but donot match to an acceptable tolerance to support the exclusive usage of direct numerical simulation for linerevaluation. However, when these impedance spectra are converted to the corresponding reflection factor magnitudeand phase, differences in the computed and measured reflection factor phase appear to be at least partly responsiblefor the impedance spectra differences. Two potential causes for these phase differences are offered. First,measurements conducted at frequencies near anti-resonance are susceptible to measurement errors, and moredetailed measurement processes may be needed. Second, the direct numerical simulation process used hereinneglects the effects of the walls of the normal incidence tube, thereby failing to account for higher-order modes thatexist in the near field of the resonator openings. Additional computations may be needed in which the entire normalincidence tube is included in the simulation.

For liners with the same porosity, it is demonstrated that the hole geometry plays a role in determining theimpedance of a liner. The absorption coefficient is found to increase with increase in the total perimeter of theresonator openings over the frequency and slit aspect ratio range studied. This is true over a wide range of soundpressure levels. It is known12, 16 that, at low incident SPL, the principal sound dissipation mechanism is viscousdissipation in the oscillatory boundary layer adjacent to the walls at the resonator openings. Thus, it is not surprisingthat the increased total perimeter (summed over all of the slits) results in increased dissipation. At high SPL, theprincipal mechanism is vortex shedding.12, 16 The acoustic energy dissipation depends on the total rotational kineticenergy imparted to the shed vortices, but the rotational kinetic energy of a vortex depends on both the length andstrength of the vortex. This results in an increase in absorption coefficient for the single slit. As the open area issubdivided into multiple slits, each slit has an increased sample thickness-to-slit width ratio, which results inincreased viscous dissipation due to the wall friction. When combined with the dominant vortex shedding, thisincreased viscous dissipation results in an increased absorption coefficient for the larger number of slits (aspect ratioof 1.25).

Finally, the present investigation extends the finding of previous studies to three dimensions that vortexshedding is the dominant sound dissipation mechanism at high sound pressure levels. However, for low aspect ratioslits, the vortex shedding process is highly regular under discrete frequency incident sound excitation. This is unlikethe irregular, fairly random vortex shedding for large aspect ratio slits (observed in two-dimensional simulations16).It is also observed that vortices, once shed and upon becoming relatively free of the influence of nearby walls, tendto readjust themselves into vortices with a circular plan form. The vortex shape readjustment and vortex dynamicalinteraction are processes that are believed to exert significant influence on the strength of shed vortices and henceliner dissipation rate. The study of vortex dynamics is, however, beyond the scope of the present work.

Acknowledgment

The work of CKWT and HJ was supported initially by a NASA Cooperative Agreement NNL04AA01A.

Appendix: Energy Conserving Discretization

In this appendix, we will discuss how to produce the pressure signal of a prescribed noise spectrum in time.Let us first show that the time varying pressure signal given by the mathematical expression below has a spectrumequal to any given spectrum S(w ) where w is the angular frequency:

p(t) = 2S(w )[ ]12 cos(wt + c(w ))dw

-•

•

Ú , (A1)

where c(w ) is a random function of w . The noise spectrum of p(t) given by Eq. (A1) is the Fourier transform

of the autocorrelation function. The autocorrelation function is,

p(t)p(t + t ) = 2 S( ¢w )S( ¢¢w )[ ]12 cos( ¢w t + c( ¢w ))cos( ¢¢w t + ¢¢w t + c( ¢¢w ))d ¢w d ¢¢w

-•

•

Ú-•

•

Ú . (A2)

The overbar in Eq. (A2) is the time average. Now,

cos( ¢w t + c( ¢w ))cos( ¢¢w t + ¢¢w t + c( ¢¢w )) = LimT Æ•

12T

cos( ¢w t + c( ¢w ))cos( ¢¢w t + ¢¢w t + c( ¢¢w ))dt-T

T

Ú

= LimT Æ•

12T

cos( ¢w t + c( ¢w )) cos( ¢¢w t + c( ¢¢w ))cos( ¢¢w t ) - sin( ¢¢w t + c( ¢¢w ))sin( ¢¢w t )[ ] }dt{-T

T

Ú . (A3)

Because c w( ) is a random function, in the limit T Æ • , the right side of Eq. (A3) is zero except when

¢w = ¢¢w . In this exceptional case, the right side of Eq. (A3) becomes 12 cos( ¢¢w t ) . Hence, it is found,

cos( ¢w t + c( ¢w ))cos( ¢¢w t + ¢¢w t + c( ¢¢w )) = 12 cos( ¢¢w t )d ( ¢w - ¢¢w ) (A4)

where d (w ) is the delta function. Substitution of Eq. (A4) into the right side of Eq. (A2) and upon integrating over

¢¢w , it is straightforward to find,

p(t)p(t + t ) = S( ¢w )cos( ¢w t )d ¢w-•

•

Ú . (A5)

Now, by taking the Fourier transform of autocorrelation function (A5), it is easy to find that the spectrum function isS(w ) . [Note: the customary assumption of S(-w ) = S(w ) is adopted].

For computation purpose, we need to discretize the integral on the right side of Eq.(A1) to produce randompressure fluctuation p(t) in real time. To this end, let us divide the spectrum into a large number of narrow bands.Let w j and Dw j be the center frequency and bandwidth of the jth band as shown in Fig. 9.

The pressure fluctuation corresponding to the energy in the jth band according to Eq. (A1) is,

pj (t) = 2S(w )[ ]12 cos(wt + c(w ))dw

w j - 12 Dw j

w j + 12 Dw j

Ú . (A6)

The energy, Ej , in this band is,

Ej = pj2 (t) = 2S( ¢w )[ ]

12 2S( ¢¢w )[ ]

12 cos( ¢w t + c( ¢w ))cos( ¢¢w t + c( ¢¢w ))Ú

w j - 12 Dw j

w j + 12 Dw j

Ú d ¢w d ¢¢w . (A7)

In the limit Dw j Æ 0 , we may take S( ¢w ) = S( ¢¢w ) = S(w j ) in the above integral. This gives,

Ej = 2S(w j ) cos( ¢w t + c( ¢w ))cos( ¢¢w t + c( ¢¢w ))Úw j - 1

2 Dw j

w j + 12 Dw j

Ú d ¢w d ¢¢w .

Upon replacing the integrand by Eq. (A4), the double integral can be evaluated to yield,

Ej = S(w j )Dw j . (A8)

Now, we will approximate the pressure fluctuations in time due to the energy in the jth band by a singleharmonic oscillation with frequency w j (center frequency of the band), i.e.,

pj (t) = 2S(w j )Dw jÈÎ ˘̊12 cos(w jt + c j ) (A9)

where c j is a random number. It is easy to show that the energy associated with pressure oscillation given by Eq.

(A9) is the same as Ej , i.e.,

pj2 (t) = 2S(w j )Dw j cos2 (w jt + c j ) = SjDw j .

Combining all the pressure fluctuations of all the spectral bands, the resulting pressure is obtained by summing overall the contributions given by Eq. (A9). This leads to the following equation for the random pressure fluctuations intime,

p(t) = 2S(w j )Dw jÈÎ ˘̊12 cos(w jt + c j )

j =1

N

 . (A10)

Equation (A10) is the energy conserving discretization of the given spectrum S(w ) .

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